Do you speak their language?

Because if everyone truly spoke the same language, wouldn't argument and disagreement be impossible?

No, not at all. Since disagreement can occur about the facts. I can speak the same language as you, but we can disagree about whether the world is round or flat, can't we? In fact, that is exactly what happened.

But can language exist based on just facts alone? .

I'll mention the tower of Babel because it is one of the more useful myths. Before the tower fell we can imagine that everyone spoke the same language and I mean truly spoke the same language. There was no disagreement about what words meant. So disagreements about what one was saying, or what it meant to say this or that, were not an issue, were not even conceivable. But this does not mean that this ancient people were not sometimes mistaken about the facts or never lied about the facts.

Aesthetic judgments offer another interesting case for disagreements.

For example: suppose Nimrod thought that blue was the most beautiful color while Abraham thought that orange was the most beautiful color. Both would be certain that each meant the same thing by calling their preferred color "the most beautiful" and both would be certain as to what each meant by "blue" and "orange". And yet they disagreed.

If everyone truly spoke the same language, wouldn't misunderstanding be impossible?

To which I would answer: Yes.

Two kinds of disagreement. Disagreement related to belief and disagreement related to attitude.

With reference to the op: Where do disagreements resulting from a lack of common language fall? (And when I say common language I don't mean like one person speaks French and the other person speaks Japanese. Hmmm... Are such conflicts really just misunderstandings? Not really disagreements at all?

So I might restate your question Render as follows:

If everyone truly spoke the same language, wouldn't misunderstanding be impossible?

To which I would answer: Yes.

Why would that be your answer? I thought you gave decisive reasons for a no answer. Why would you think that if one person thinks that the capital of Ecuador is Quito, and another that the capital is La Paz, that would just be a misunderstanding? It would be a clear disagreement of belief (fact). One would be right, and the other wrong. End of story.

You are correct. "Misunderstanding" is not specific enough and simply the wrong word to use here.

So I will restate Render's question again:

If everyone truly spoke the same language, wouldn't mistranslation be impossible?

I say: Yes

There is almost a need for another word. "Mistranslation" denotes usually only a mistranslation between established languages like Spanish and Russian and only rarely and almost metaphorically does "mistranslation" denote mistranslation within an established language such as happens between different jargons and slangs and more minute linguistic differences at the intrapersonal level. Nevertheless "mistranslation" is greatly preferable to "misunderstanding".

It does appear there be need for another word in this circumstance. If everyone truly spoke the same language there would be no translation, of course the problem would then be one of interpretation.
Translaters were once refered to as interpreters, in which case a misinterpretation would result in a mistranslation, which would occur at the level of slang and jargon as well.

Yes! "Misinterpretation" is the better word.

If everyone truly spoke the same language/jargon/slang, wouldn't misinterpretation would be impossible?

To which I answer: Yes

I'm not so sure it would be impossible, doesn't interpretation depend a lot on our perspective and perceptions?
Of course I am not clear exactly what is to be meant by truly speaking the same language, is that to mean a universal understanding included?

And if that is so, wouldn't that mean we would all need to have the same experiences from the same perspective?

Our understanding of language is acquired through experience and this experience affects language denotatively and conotatively but I think the spirit of the OP is that all such differences would be eliminated though we all may very well have very different experiences.

Suppose we all have input jacks in the backs of our head and download the exact same language fed directly into the language centers of our brains... perhaps the dictionary is constantly updated via a wireless internet connection... but this is stopping well short of a hive mind where everyone can hear everyone's thoughts... just the words mean the same thing for everyone... no one knows what anyone else is thinking until words are actually spoken out loud.

You are correct. "Misunderstanding" is not specific enough and simply the wrong word to use here.

So I will restate Render's question again:

If everyone truly spoke the same language, wouldn't mistranslation be impossible?

I say: Yes

There is almost a need for another word. "Mistranslation" denotes usually only a mistranslation between established languages like Spanish and Russian and only rarely and almost metaphorically does "mistranslation" denote mistranslation within an established language such as happens between different jargons and slangs and more minute linguistic differences at the intrapersonal level. Nevertheless "mistranslation" is greatly preferable to "misunderstanding".

Current Cognitive Linguisitc studies demonstrate that vocabulary is processed with a prototypical schema, that has categories and subcategories that overlap with other prototypical schemas with their categopries and subcategories. Every referent has a generally accepted prototype (thing-ness) or (attribute-ness) or (action-ness) whatever. Categories and subcategories can be hierarchically more or less specific than the prototype. So what we have visually represented, imagine a super complex ven diagram with a large prototype circle and various categorical circles inside it overlapping each other and extending past the prototypical circle into other large protypical circles etc... It would be highly improbable that every word spoken by one would fall in the same place on the ven diagram as the word interpreted by the listener. This does not mean that communication is futile, after all both parties still process the (thing-ness) of the word, however in complex arguments it becomes ever more important to pinpoint the vocabulary to a fixed position using modifiers and such to keep from having misunderstandings. The problem is that humans just don't have the time and normally don;t have the need to take the effort to pinpoint these things.

Because if everyone truly spoke the same language, wouldn't argument and disagreement be impossible?

So why do some people try so hard to convince other people of their philosophical idea's when their concepts will always be interpreted differently than they had intended?